I am just curious. Based on your personal experience, is Rice well-known on the east and west coasts? How prestigious is it compared to the ivy league schools?
Rice is one of the best schools in the country and any grad school or hiring manager will know that. Might your aunt Millie or your friend’s dad? Who cares.
It’s especially well known for engineering at major corps and grad schools and better at those than most Ivy’s. It also has the advantage of internships with its location in downtown Houston.
We live in an SoCal community that has been included on some lists (e.g. GreatSchools.org) as having one of the best public education systems in the country. Many wealthy immigrants settle here in no small part because of the good schools, and the high school curriculum caters to their kids, many of whom dream of winning admission to a top US university. Though my daughter–who is currently at Rice–attended a charter high school elsewhere in our county, she has been surprised at how many other students from her hometown have ended up at Rice. I’d say the fact that so many strong graduates from a single town’s public school system in a prime region of the nation have trekked halfway across the continent to attend a small university in Houston attests to the solid reputation of that university among bright kids on the west coast.
However, precisely because Rice is small (and is located in the relatively new and unglamorous metropolis of Houston rather than in, say, Boston or NYC or Chicago or the Bay Area), fewer people–both among the general public and among those applying to colleges–will be familiar with Rice, its unique features, its reputation, and its rankings.
Of course, as others have noted, grad schools and many top employers will be familiar with Rice. For example, during the fall recruiting season this year, many investment banks (both bulge brackets and boutiques) are targeting Rice, as are most of the major management consulting firms, not to mention an array of tech companies, from Facebook and Apple to various start-ups. This bespeaks a school that has prestige where it counts. On the other hand, if you crave bragging rights with your non-college-nerd friends, you might well be disappointed. Rice lags behind the Ivies and such Ivy equivalents as Chicago and Duke in popular prestige, though I think it compares favorably to other peer schools like WashU or Emory.
For relevant people, Rice is very well regarded.
For irrelevant people, why do you care?
Ditto. Rice will be known by people that matter. Same is true for other small, excellent schools like Harvey Mudd.
Rice is the top university in Texas. Every city in Texas recognizes Rice as being a very selective and “smart” school. So it is pretty well known, but mostly in Texas and surrounding states. 
Rice is definitely not known well in the North East. If people have heard of it, they don’t know how good of a school it is
Yes among “lay people” it isn’t necessarily well known outside of it’s region.
Who cares, those who matter will know.
I live in New England and am applying to Rice, but I have not encountered many people at all who have heard of it, unless they are high school seniors who have recently looked at rankings of top colleges.
I’m applying from the Midwest, and it seems to be relatively unknown here. While I think it’s more common to know about its status closer to Texas, in the Midwest and Northeast, the Ivies (and colleges like Duke, Stanford, MIT, etc) are usually more well-known.
I don’t think this is a problem unique to Rice. There are top schools which most people know (Harvard, MIT, Stanford), and there are schools well-known because of sports or due to being big state schools etc. Besides those, the majority of really good schools just don’t seem to be well-known by people outside of their specific region. It’s understandable why: they’re small and selective and attract only a very academic bunch, a group quite unlike most of the population. Here in the South, for example, few people seem to know places like Brown or Northwestern, and LACs like Amherst and Williams are even lesser known.
But when it comes to people who matter, like employers and grad schools, it usually is not an issue 
It’s not on the radar for average people in the San Francisco Bay Area, and students out here rarely want to go to Texas.
But I guarantee that every hiring manager for tech jobs in Silicon Valley and San Francisco knows it quite well.