It should be reasonably simple to create a database program like this, but that doesn't mean there isn't a lot of effort involved. There are four parts to it -
Designing the database. Working out how your data is stored in the EEPROM and how it will be accessed. A rough guesstimate, with a 2x16 LCD x 2000 plants that's a 64KB/512Kbit EEPROM. Holding less data means a smaller EEPROM.
Initialising the database. A PC database sent serially to a program dedicated to moving that into EEPROM probably.
Updating the database on your travels. A program to get a data record from the EEPROM, put it to the LCD, update any related numerical data, and put that back into EEPROM.
Updating the PC. Running through all the records and sending the data to the PC.
Each is relatively straight forward in itself and can be made more functional as time goes by.
The two issues you may need to think on is how you are going to do the PC side of things, to create the database and get the data into the EEPROM. And, how to efficiently update EEPROM without having to re-write whole blocks of data each time. I may be overly worrying about that. Two databases might be an idea; one with plant names, the other with updatable data. I guess KEYIN could be used so you could create the database on the PICAXE using a PC keyboard and not have to use a PC there at all.
On the hardware side you'll need a fairly battery efficient design, perhaps using rechargables, and a mechanism to warn of battery low and avoid corrupting EEPROM.
A PDA might be the better course; you can get an old Palm Pilot with nice LCD, graphical interface and touch screen for the same price as an 18X chip or a 2x16 LCD, but you would still have all the design and programming issues you'd have with a PICAXE.
Psion Organisers may be a good bet, because they have OPL, a programming language which is also BASIC-like, built-in and handles databases like this. I have a very old Psion II which I use as a telephone number book. It ran for three years on one PP3 battery and kept almost second-accurate time long term !