Paul, who's running for the GOP nomination for president, has said he won't seek re-election next year.
Weber announced an exploratory effort in July and says he found the support he needed. He's been in the Texas House since 2009 and served on the Pearland City Council for six years before that.
That looks to be a busy primary. Felicia Harris, an attorney and a current Pearland City Council member, is running in the GOP primary, and so is Michael Truncale of Jefferson County, a regent at the Texas State University System and a member of the State Republican Executive Committee.
Up in North Texas, state Rep. Rodney Anderson, R-Grand Prairie, has said on paper what he's been saying on the phone and in person for weeks: He plans to run for state Senate against incumbent Republican Sen. Chris Harris of Arlington. Anderson won his House seat in November, knocking off incumbent Democrat Kirk England in a squeaker.